# Entropy, convergence and invariant measures

Could you give conditions that a sequence of shift invariant measures $\eta_{n}$ has to satisfy in order to happen this convergence in terms of entropies $h(\eta_{n})$:

$h(\eta_{n}) \rightarrow \log2$?

(here I'm thinking about the Bernoulli space $\{0,1\}^{\mathbb{N}}$ and usual shift function on this space)

I'm in trouble with this...

Thanks again

• There is only one measure of maximal entropy in this case... – Asaf Jul 19 '15 at 20:30
• The standard condition for this to happen is $\bar d(\eta_n,\mu)\to 0$, where $\mu$ is the $(\frac 12,\frac 12)$ Bernoulli measure. The distance $\bar d$ is called the "d-bar" distance. It is described (for example) in Rudolph's book. One definition of $\bar d(\mu,\nu)$ is that is the limit of the minimal integral of $\frac 1n\sum_{i=0}^{n-1}\Delta(x_i,y_i)$ over all couplings of $\mu$ and $\nu$. – Anthony Quas Jul 19 '15 at 22:41